\(\int \frac {A+B x^2}{(d+e x^2) \sqrt {a d^2-a e^2 x^4}} \, dx\) [3]

Optimal result

Integrand size = 36, antiderivative size = 187 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=-\frac {(B d-A e) x \sqrt {a d^2-a e^2 x^4}}{2 a d^2 e \left (d+e x^2\right )}-\frac {(B d-A e) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} e^{3/2} \sqrt {a d^2-a e^2 x^4}}+\frac {B \sqrt {d} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{e^{3/2} \sqrt {a d^2-a e^2 x^4}} \] Output:

-1/2*(-A*e+B*d)*x*(-a*e^2*x^4+a*d^2)^(1/2)/a/d^2/e/(e*x^2+d)-1/2*(-A*e+B*d 
)*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(3/2)/(-a 
*e^2*x^4+a*d^2)^(1/2)+B*d^(1/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/ 
d^(1/2),I)/e^(3/2)/(-a*e^2*x^4+a*d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.60 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=-\frac {-\sqrt {-\frac {e}{d}} (B d-A e) x \left (d-e x^2\right )+i d (B d-A e) \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 i B d^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{2 d^3 \left (-\frac {e}{d}\right )^{3/2} \sqrt {a \left (d^2-e^2 x^4\right )}} \] Input:

Integrate[(A + B*x^2)/((d + e*x^2)*Sqrt[a*d^2 - a*e^2*x^4]),x]
 

Output:

-1/2*(-(Sqrt[-(e/d)]*(B*d - A*e)*x*(d - e*x^2)) + I*d*(B*d - A*e)*Sqrt[1 - 
 (e^2*x^4)/d^2]*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - (2*I)*B*d^2*Sqr 
t[1 - (e^2*x^4)/d^2]*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1])/(d^3*(-(e/d 
))^(3/2)*Sqrt[a*(d^2 - e^2*x^4)])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1396, 402, 25, 27, 399, 289, 329, 327, 765, 762}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx\)

\(\Big \downarrow \) 1396

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \int \frac {B x^2+A}{\left (e x^2+d\right )^{3/2} \sqrt {a d-a e x^2}}dx}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (-\frac {\int -\frac {a \left (d (B d+A e)-e (B d-A e) x^2\right )}{\sqrt {e x^2+d} \sqrt {a d-a e x^2}}dx}{2 a d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\int \frac {a \left (d (B d+A e)-e (B d-A e) x^2\right )}{\sqrt {e x^2+d} \sqrt {a d-a e x^2}}dx}{2 a d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\int \frac {d (B d+A e)-e (B d-A e) x^2}{\sqrt {e x^2+d} \sqrt {a d-a e x^2}}dx}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {2 B d^2 \int \frac {1}{\sqrt {e x^2+d} \sqrt {a d-a e x^2}}dx-(B d-A e) \int \frac {\sqrt {e x^2+d}}{\sqrt {a d-a e x^2}}dx}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\frac {2 B d^2 \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d+e x^2} \sqrt {a d-a e x^2}}-(B d-A e) \int \frac {\sqrt {e x^2+d}}{\sqrt {a d-a e x^2}}dx}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\frac {2 B d^2 \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d+e x^2} \sqrt {a d-a e x^2}}-\frac {d \sqrt {1-\frac {e^2 x^4}{d^2}} (B d-A e) \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d+e x^2} \sqrt {a d-a e x^2}}}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\frac {2 B d^2 \sqrt {a d^2-a e^2 x^4} \int \frac {1}{\sqrt {a d^2-a e^2 x^4}}dx}{\sqrt {d+e x^2} \sqrt {a d-a e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (B d-A e) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d+e x^2} \sqrt {a d-a e x^2}}}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\frac {2 B d^2 \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d+e x^2} \sqrt {a d-a e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (B d-A e) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d+e x^2} \sqrt {a d-a e x^2}}}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d+e x^2} \sqrt {a d-a e x^2} \left (\frac {\frac {2 B d^{5/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d+e x^2} \sqrt {a d-a e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} (B d-A e) E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d+e x^2} \sqrt {a d-a e x^2}}}{2 d^2 e}-\frac {x \sqrt {a d-a e x^2} (B d-A e)}{2 a d^2 e \sqrt {d+e x^2}}\right )}{\sqrt {a d^2-a e^2 x^4}}\)

Input:

Int[(A + B*x^2)/((d + e*x^2)*Sqrt[a*d^2 - a*e^2*x^4]),x]
 

Output:

(Sqrt[d + e*x^2]*Sqrt[a*d - a*e*x^2]*(-1/2*((B*d - A*e)*x*Sqrt[a*d - a*e*x 
^2])/(a*d^2*e*Sqrt[d + e*x^2]) + (-((d^(3/2)*(B*d - A*e)*Sqrt[1 - (e^2*x^4 
)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d + e*x^2 
]*Sqrt[a*d - a*e*x^2])) + (2*B*d^(5/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[A 
rcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d + e*x^2]*Sqrt[a*d - a*e*x 
^2]))/(2*d^2*e)))/Sqrt[a*d^2 - a*e^2*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d 
+ c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, 
x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.27

method	result	size
elliptic	\(\frac {\left (-a \,e^{2} x^{2}+a d e \right ) x \left (A e -B d \right )}{2 d^{2} a \,e^{2} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}+a d e \right )}}+\frac {\left (\frac {B}{e}+\frac {A e -B d}{2 d e}\right ) \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}-\frac {\left (A e -B d \right ) \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}\, e}\)	\(238\)
default	\(\frac {B \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{e \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}+\frac {\left (A e -B d \right ) \left (\frac {\left (-a \,e^{2} x^{2}+a d e \right ) x}{2 d^{2} a e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-a \,e^{2} x^{2}+a d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-a \,e^{2} x^{4}+a \,d^{2}}}\right )}{e}\)	\(280\)

Input:

int((B*x^2+A)/(e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/2*(-a*e^2*x^2+a*d*e)/d^2*x/a*(A*e-B*d)/e^2/((x^2+d/e)*(-a*e^2*x^2+a*d*e) 
)^(1/2)+(B/e+1/2/d/e*(A*e-B*d))/(1/d*e)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d 
)^(1/2)/(-a*e^2*x^4+a*d^2)^(1/2)*EllipticF(x*(1/d*e)^(1/2),I)-1/2*(A*e-B*d 
)/d/(1/d*e)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-a*e^2*x^4+a*d^2)^( 
1/2)/e*(EllipticF(x*(1/d*e)^(1/2),I)-EllipticE(x*(1/d*e)^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=-\frac {{\left (B d^{2} e - A d e^{2} + {\left (B d e^{2} - A e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left (B d^{3} + {\left (A + B\right )} d^{2} e - A d e^{2} + {\left (B d^{2} e + {\left (A + B\right )} d e^{2} - A e^{3}\right )} x^{2}\right )} \sqrt {a d^{2}} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + \sqrt {-a e^{2} x^{4} + a d^{2}} {\left (B d^{2} e - A d e^{2}\right )} x}{2 \, {\left (a d^{3} e^{3} x^{2} + a d^{4} e^{2}\right )}} \] Input:

integrate((B*x^2+A)/(e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="frica 
s")
 

Output:

-1/2*((B*d^2*e - A*d*e^2 + (B*d*e^2 - A*e^3)*x^2)*sqrt(a*d^2)*sqrt(e/d)*el 
liptic_e(arcsin(x*sqrt(e/d)), -1) - (B*d^3 + (A + B)*d^2*e - A*d*e^2 + (B* 
d^2*e + (A + B)*d*e^2 - A*e^3)*x^2)*sqrt(a*d^2)*sqrt(e/d)*elliptic_f(arcsi 
n(x*sqrt(e/d)), -1) + sqrt(-a*e^2*x^4 + a*d^2)*(B*d^2*e - A*d*e^2)*x)/(a*d 
^3*e^3*x^2 + a*d^4*e^2)
 

Sympy [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\int \frac {A + B x^{2}}{\sqrt {- a \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )} \left (d + e x^{2}\right )}\, dx \] Input:

integrate((B*x**2+A)/(e*x**2+d)/(-a*e**2*x**4+a*d**2)**(1/2),x)
 

Output:

Integral((A + B*x**2)/(sqrt(-a*(-d + e*x**2)*(d + e*x**2))*(d + e*x**2)), 
x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x^2+A)/(e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="maxim 
a")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {-a e^{2} x^{4} + a d^{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x, algorithm="giac" 
)
 

Output:

integrate((B*x^2 + A)/(sqrt(-a*e^2*x^4 + a*d^2)*(e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {a\,d^2-a\,e^2\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:

int((A + B*x^2)/((a*d^2 - a*e^2*x^4)^(1/2)*(d + e*x^2)),x)
 

Output:

int((A + B*x^2)/((a*d^2 - a*e^2*x^4)^(1/2)*(d + e*x^2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \sqrt {a d^2-a e^2 x^4}} \, dx=\frac {\sqrt {a}\, \left (-\sqrt {-e^{2} x^{4}+d^{2}}\, b x +2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) a \,d^{2} e +2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \right ) a d \,e^{2} x^{2}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e \,x^{2}+d}d x \right ) b d +\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e \,x^{2}+d}d x \right ) b e \,x^{2}\right )}{2 a d e \left (e \,x^{2}+d \right )} \] Input:

int((B*x^2+A)/(e*x^2+d)/(-a*e^2*x^4+a*d^2)^(1/2),x)
 

Output:

(sqrt(a)*( - sqrt(d**2 - e**2*x**4)*b*x + 2*int(sqrt(d**2 - e**2*x**4)/(d* 
*3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)*a*d**2*e + 2*int(sqrt(d**2 
- e**2*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)*a*d*e**2*x* 
*2 + int(sqrt(d**2 - e**2*x**4)/(d + e*x**2),x)*b*d + int(sqrt(d**2 - e**2 
*x**4)/(d + e*x**2),x)*b*e*x**2))/(2*a*d*e*(d + e*x**2))